Quantum control of qubits and atomic motion using ...

Quantum control of qubits and atomic motion using ultrafast laserpulses

J. Mizrahi • B. Neyenhuis • K. G. Johnson •

W. C. Campbell • C. Senko • D. Hayes •

C. Monroe

Received: 1 July 2013 / Accepted: 25 October 2013

� Springer-Verlag Berlin Heidelberg 2013

Abstract Pulsed lasers offer significant advantages over

continuous wave (CW) lasers in the coherent control of

qubits. Here we review the theoretical and experimental

aspects of controlling the internal and external states of

individual trapped atoms with pulse trains. Two distinct

regimes of laser intensity are identified. When the pulses

are sufficiently weak that the Rabi frequency X is much

smaller than the trap frequency xtrap, sideband transitions

can be addressed and atom-atom entanglement can be

accomplished in much the same way as with CW lasers. By

contrast, if the pulses are very strong X� xtrap, impulsive

spin-dependent kicks can be combined to create entangling

gates which are much faster than a trap period. These fast

entangling gates should work outside of the Lamb-Dicke

regime and be insensitive to thermal atomic motion.

1 Introduction

Over the past decade, frequency combs from mode-locked

lasers have become an essential tool in the field of optical

frequency metrology [1–4]. This is due to the broad spec-

trum of lines spaced by the pulse repetition rate present in a

frequency comb. This allows it to serve as a precise con-

nection between distant frequencies. In the context of

metrology, this feature is used as a ruler in which the

spacings between comb lines serve as tick marks. In the

context of coherent control, this feature can be used to

directly bridge large frequency gaps between energy levels

in a controllable way. This technique has been used

effectively to control diverse quantum systems, including

multilevel atoms [5], molecules [6], semiconductor spin

states [7, 8], and ions [9–11]. Mode-locked lasers therefore

have a bright future as a tool for qubit manipulation in a

number of different quantum computer architectures.

Trapped atomic ions are a very promising medium for

quantum information, due to their extremely long coher-

ence times, well-established means for coherent control

and manipulation, and potential for scalability [12, 13].

High-fidelity entanglement of ions is now routinely

achieved [14–17], as well as implementations of schemes

for analog quantum simulation [18–20] and digital quan-

tum algorithms [21–23]. However, obstacles remain before

a trapped ion quantum computer can outperform a classical

computer. Technical limitations to gate fidelity include

laser-induced decoherence [24, 25] and ion heating [26].

Existing gates are also typically limited in the number of

ions which can be manipulated in a single chain. This is

because these gates rely on addressing normal modes of

motion of the ion chain [27, 28]. As the number of ions

grows, the density of normal modes in frequency space

grows as well, making it increasingly difficult to avoid

undesired couplings. This increased mode density slows

down the gate, increasing sensitivity to low-frequency

noise.

High-power mode-locked lasers offer one potential

solution to some of these issues (there are a number of

other approaches, see [29–32]). The goal of this paper is to

discuss recent work on the interaction between trapped ions

and mode-locked laser pulses.

J. Mizrahi (&) � B. Neyenhuis � K. G. Johnson � C. Senko �D. Hayes � C. Monroe

Department of Physics and National Institute of Standards

and Technology, Joint Quantum Institute, University

of Maryland, College Park, College Park, MD 20742, USA

e-mail: [email protected]

W. C. Campbell

Department of Physics and Astronomy, University of California,

Los Angeles, Los Angeles, CA 90095, USA

123

Appl. Phys. B

DOI 10.1007/s00340-013-5717-6

From a technical standpoint, the large bandwidth inher-

ent in a comb eliminates some of the complexity and

expense of driving Raman transitions. For hyperfine qubits

in ions, the frequency splitting is typically several GHz.

Bridging this gap with CW beams requires either two sep-

arate phase-locked lasers or a high-frequency EOM (which

is typically inefficient). By contrast, a single mode-locked

laser can directly drive the transition without any high-

frequency shifts. Moreover, it is not necessary to stabilize

either the carrier-envelope phase or the repetition rate of the

mode-locked laser, as will be discussed later. This enables

the use of commercially available, industrial lasers.

As a second advantage, the large instantaneous intensity

present in a single pulse enables efficient harmonic gen-

eration. For this reason, high-power UV lasers are readily

obtainable at frequencies appropriate for trapped ion con-

trol. High power enables operating with a large detuning,

which suppresses laser-induced decoherence. High power

also enables motion control in a time significantly faster

than the trap period, which is a new regime in trapped ion

control. It should allow the implementation of theoretical

proposals for ultrafast gates which are independent of ion

temperature, as discussed in Sect. 4.

This paper is divided into three parts. Section 2

describes spin control of an ion with a pulse train, without

motional coupling. Section 3 introduces spin–motion cou-

pling. Section 4 explains how to realize an ultrafast two ion

gate using fast pulses.

1.1 Experimental system

We take the atomic qubit as composed of stable ground-

state electronic levels separated by RF or microwave fre-

quencies. The schemes reported here can be extended to

the case of qubit levels separated by optical intervals, but

for concreteness we will concentrate on qubits stored in

hyperfine or Zeeman levels in the ground state of an alkali-

like atom.

In order to effectively use laser pulses for qubit control,

we require three frequency scales to be well separated. Let sdenote the pulse duration. The pulse bandwidth 1/s should

be much larger than the qubit splitting xq so that the two

qubit levels are coupled by the optical field, yet it should be

much smaller than the detuning D from the excited state so

that it is negligibly populated during the interaction. Note

also that the detuning D should not be much larger than the

fine structure splitting in an alkali-like atom; otherwise, the

Raman coupling is suppressed [16]. For many atomic sys-

tems, the condition xq � 1=s� D is satisfied for a range

of laser pulse durations 0.5 ps [ s [ 25 ps.

Here we consider the interaction between ultrafast laser

pulses and qubits represented by laser-cooled 171Yb? ions

confined in an RF Paul trap, although many of the results

discussed herein are applicable in a range of contexts

involving ultrafast pulses on the internal and external

degrees of freedom of optically coupled qubits. The qubit

levels are defined by the mF = 0 states of the 2S1/2

hyperfine manifold of 171Ybþ : jF ¼ 0; mF ¼ 0i� j0i; jF ¼ 1; mF ¼ 0i � j1i. The qubit frequency split-

ting is xq/2p = 12.6 GHz. Doppler cooling of atomic

motion, and initialization/detection of the qubit are all

accomplished using continuous wave (CW) beams near

369 nm [33].

We consider optical pulses generated from a mode-

locked, tripled, Nd:YVO4 laser at 355 nm to drive stimu-

lated Raman transitions between the qubit states j0i and

j1i; that may also be accompanied by optical dipole forces.

Typical laser repetition rates are in the range xrep/

2P1/2

2S1/2 ωq/2π = 12.6 GHz

2P3/2

370 nm

355

nm

33 THz67 THz

|0⟩|1⟩

σ+σ−

Fig. 1 Relevant energy levels of 171Yb?. The qubit is identified with

the two mF = 0 states in the ground-state manifold. Continuous wave

369 nm light is used for cooling, detection, and optical pumping.

Laser pulses at 355 nm are used for qubit manipulation, driving

stimulated Raman transitions between the qubit levels from r±

polarized light

320 340 360 380

Wavelength [nm]

Spo

n. E

mis

s. P

roba

bilit

y

1x10-5

2x10-5

0

|ΔE

Light / eff |

1x10-3

2x10-3

0

Fig. 2 Theoretical curves showing sources of laser-induced deco-

herence as a function of wavelength. Solid blue line is spontaneous

emission probability during a p pulse as a function of laser

wavelength. Dashed red line is differential AC Stark shift divided

by Rabi frequency as a function of laser wavelength. White squares

are at 355 nm, where both curves are near a minimum

J. Mizrahi et al.

123

2p = 80–120 MHz, with a pulse duration s * 10 ps

(*100 GHz bandwidth) and maximum average power �P of

several Watts (pulse energies of up to 100 nJ). This light is

detuned by D1=2 � þ33 THz from the excited 2P1/2 level,

and D3=2 � �67 THz from the 2P3/2 level, as shown in

Fig. 1. This wavelength and pulse duration are nearly

optimal for controlling the 171Yb? system, exhibiting

minimal spontaneous emission and differential AC Stark

shifts [10], as shown in Fig. 2.

2 Spin control with pulses

2.1 Strong pulses

Consider the interaction of a train of pulses with an atom,

as shown in Fig. 3. After performing a rotating wave

approximation at the optical frequency and adiabatically

eliminating the excited P states, the effective Hamiltonian

for the interaction becomes [10]:

H ¼ �xq

2r̂z �

XðtÞ2

r̂x ð1Þ

where xq is the qubit splitting, r̂z;x are Pauli spin operators,

and XðtÞ ¼ gðtÞ2=2D is the two-photon Rabi frequency for

pure r? or r- polarized light. Here, the single-photon S -

P resonant coupling strength gðtÞ ¼ cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

IðtÞ=2Isat

p

with

effective detuning given by 1=D ¼ 1=D1=2 � 1=D3=2,

accounting for both excited states. I(t) is time-dependent

intensity of the pulse. In the 171Yb? system, Isat = 0.15

W/cm2 is the saturation intensity for the 2S1/2 - 2P1/2

transition and the 2P1/2 state linewidth is c/2p = 19.6 MHz.

We note that the above Hamiltonian can be generalized

to include the effect of ultrafast pulses connecting the qubit

levels to a third (transiently populated) level on resonance,

or in the case of qubits with an optical splitting, directly on

resonance with the qubit levels [34]. In addition, by

choosing appropriate qubit levels and laser pulse polariza-

tion, a generalization of the above interaction can produce a

differential Stark shift instead of a transition between the

levels, in which case the r̂x coupling term above is replaced

by r̂z [34]. In this case, the actual implementation of

entangling gates between multiple qubits through collective

motion is not exactly as described below, although there are

many similarities. It should also be noted that qubit states

that have sizable differential AC Stark shift are also first-

order sensitive to external magnetic fields [16] and hence

perform as relatively poor qubit memories.

For a single pulse (N = 1) with either r± polarization,

the time dependence of the Rabi frequency originates from

the intensity profile of the laser I(t), which for a mode-

locked laser pulse can be accurately modelled by a squared

hyperbolic secant envelope [35]. Intensity envelope func-

tions of externally generated optical harmonics of the

fundamental laser field should be higher powers of the sech

function. However, their shape remains quite similar to that

of the sech function. We therefore approximate the pulse

intensity as IðtÞ ¼ I0 sech pts

� �

with peak laser intensity I0

and pulse duration s, having FWHM in time of 0.838s.

This approximation allows a simple analytic solution to the

evolution of the above Hamiltonian, and numerical simu-

lation indicates that this is at most a 1–2 % correction to

everything presented here.

The qubit Rabi frequency can therefore be written as:

XðtÞ ¼ hs

sechpt

s

� �

; ð2Þ

where h ¼R

XðtÞdt is the pulse area. For the Raman

transition considered here in the 171Yb? system using light

tuned to 355 nm, we have [9]:

h ¼ I0sc2

2IsatDð3Þ

Alternatively, h can be expressed in terms of the average

intensity of the laser �I and the repetition frequency xrep

using the relation I0s ¼ 2p�I=xrep. We find that to drive a

Raman p-pulse with a single laser pulse focussed to a

Gaussian waist w (1/e field radius), the required pulse

energy is Ep ¼ pI0w2s=2 ¼ p2Isatw2D=c2. For the 171Yb?

system using a 355-nm beam focused to a waist of

w = 10 lm, we find Ep � 12 nJ.

The Hamiltonian of Eqs. 1 and 2 for the hyperbolic

secant Rabi frequency envelope in time was solved exactly

by Rosen and Zener [36]. For the purposes of this analysis,

we are not interested in the dynamics during the pulse, but

only the resultant state after the pulse. The evolution

operator for a pulse followed by free evolution for a time

T is given by [37, 38]:

U ¼ A iB

iB A

� �

ð4Þ

where A and B are given by:

A ¼ C2 nð ÞeixqT=2

C n� h2p

� �

C nþ h2p

� � ð5Þ

B ¼ � sinh2

� �

sechxqs

2

� �

e�ixqT=2 ð6Þ

mode-locked355 nm laser

pulsepicker

Fig. 3 A fast pulse picker selects a train of N circularly polarized

pulses, each with area h. These pulses drive stimulated Raman

transitions in a trapped ion

Quantum control of qubits and atomic motion

123

n ¼ 1

2þ i

xqs2p

ð7Þ

where CðnÞ is the gamma function. For a fixed value of h,

this evolution operator can be written as a pure rotation

operator:

~U ¼ eiun̂�r~=2 ð8Þ

where the rotation axis n̂ and rotation angle u are given by:

cosu2

� �

¼ Re Að Þ ð9Þ

nz sinu2

� �

¼ Im Að Þ ð10Þ

ðnx þ inyÞ sinu2

� �

¼ B ð11Þ

The equivalent pure Bloch sphere rotation is shown in

Fig. 4b. Equation 8 allows the evolution operator to

quickly be extended to N pulses equally spaced by a time T:

UN ¼ eiNun̂�r~=2 ð12Þ

If the ion is initialized to the state j0i, then the transition

probability after N pulses is given by:

P0!1 ¼ i sinNu2

� �

nx þ iny

� �

2

¼ jBj2

sin2 u2

� �

!

sin2 Nu2

� �

ð13Þ

To understand the behavior described by the above

equations, first consider the limit of an infinitesimally short

pulse: s = 0. In that case, Eqs. 5 and 6 become:

A ¼ cosh2

� �

eixqT=2 ð14Þ

B ¼ � sinh2

� �

eixqT=2 ð15Þ

If the time between pulses satisfies the condition:

xqT ¼ 2pn; n 2 Z ð16Þ

then Eqs. 9, 10, and 11 show that u ¼ h, nz = ny = 0, and

nx = 1. In this case, the action of each pulse is rotation

about the x-axis, by an angle equal to the pulse area.

Equation 13 then becomes:

P0!1 ¼ sin2 Nh2

� �

ð17Þ

This equation shows that the behavior is discretized Rabi

flopping.

Now consider nonzero pulse duration. Equation 13

shows that for N = 1, the transition probability reduces to:

P0!1 ¼ jBj2 ¼ sin2ðh=2Þsech2ðxqs=2Þ ð18Þ

Therefore, for a single pulse, the maximum population

transferred is sech2(xqs/2). This quantity is always less

than one, meaning a single pulse cannot fully flip the spin

of the qubit. However, for two pulses, Eq. 13 can be made

equal to 1, for particular values of the delay time T. If T�1/xq, then the correct delay condition will be a small

correction to Eq. 16.

This can be understood by examining the qubit evolu-

tion on the Bloch sphere. The Bloch sphere path for the

Rosen-Zener solution is shown as a function of h in

Fig. 4a. Note that the path is twisted—for small values of

h, the rotation axis is nearly purely about the x-axis. As hincreases, the amount of z-rotation also increases. If h is

fixed, the final state can be connected to the initial state by

a pure rotation, which is shown in Fig. 4b. For nonzero

pulse duration, the rotation axis is never purely in the x–

y plane, meaning the north pole of the Bloch sphere is

never reached. However, two pulses can fully flip the spin,

so long as one pulse can reach the equator, as shown in Fig.

4e. For 171Yb?, the condition for two pulses to be able to

fully transfer population is s\ 22 ps.

These results show that two fast pulses can be used to

rotate the state of a qubit extremely rapidly, in less than one

qubit period. Moreover, these same pulses can be used to

rotate the phase of a qubit (i.e., z-rotations on the Bloch

sphere). To see this, again consider a pair of pulses as

above. However, instead of choosing a delay such that

Eq. 13 equals 1, a delay is chosen such that it equals 0; i.e.,

u ¼ p. In that case, the evolution operator causes a phase

shift of the qubit, controllable via the power of the pulses.

Figure 4d shows experimental results for a single pulse.

The datasets shown correspond to two different lasers with

different pulse durations. The circles shows a maximum

brightness of 72 %, corresponding to a pulse duration of

s = 14.8 ps in Eq. 18. The squares shows a maximum of

91 %, corresponding to s = 7.6 ps. These numbers are

consistent with independent measurements of the pulse

duration.

Figure 4f shows the results of scanning the delay

between two pulses. The two pulses were created by

splitting a single pulse from the laser and directing the

two halves of the pulse onto the ion from opposite

directions, as described in [10]. (Note that while the pulses

are directed onto the ion from opposite directions, there is

no coupling to the ion’s motion—the pulses are not

overlapped in time. There is therefore no possibility of

momentum transfer.) The maximum occurs at a delay of

72 ps, approximately one qubit period. It is slightly less

than one qubit period due to the off-axis rotation caused

by the nonzero pulse duration, as shown on the Bloch

sphere in Fig. 4e. The maximum is less than one due to

detection errors.

J. Mizrahi et al.

123

To demonstrate pure phase rotation, the delay between

the pulses was set such that there was no net population

transfer (34 ps delay in Fig. 4f). This pulse pair was then

put between two p/2 Ramsey zones, and the frequency of

those Ramsey zones scanned for different laser intensities.

The phase shift caused by the laser pulses manifests as a

shift in the Ramsey fringes. The angle of z-rotation can

then be calculated based on the shift. The amount of phase

rotation is set by controlling the intensity of the two pulses.

The results are shown in Fig. 5. The fit curve in (c) is

derived from the Rosen-Zener solution, Eqs. 5–7. The only

free parameter is the overall scaling, i.e., the

(a)

(e)1

0

20 30 40 50 60 70 80 90

Delay Between Pair of Pulses (ps)

Tra

nsiti

on P

roba

bilit

y

0.0

0.2

0.4

0.6

0.8

1.0

(c)

(b)

(d)

Max

imum

Sin

gle

Pul

seP

opul

atio

n T

rans

fer

0.0

0.2

0.4

0.6

0.8

1.0

20 30 40 500 10Pulse Duration (ps)

Tra

nsiti

on P

roba

bilit

y

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15Pulse Energy (arb. units)

(f)

0

1

0

Fig. 4 a Bloch sphere position as a function of pulse energy,

following the Rosen–Zener solution in Eqs. 5–7. b The final position

reached by the twisted path in a can be represented by a single

effective rotation axis and angle, as in Eq. 8. The angle of rotation is

given by u; the axis is determined by h and s. c Theoretical maximum

population transfer in 171Yb? for a single pulse as a function of pulse

duration, based on Eq. 18. The black dots indicate the points

corresponding to the data in (d). d Experimental data showing the

behavior described theoretically in (a–c). Ion state is measured as a

function of incident pulse energy. The transfer probability reaches a

maximum given by Eq. 18. The two different datasets correspond to

two different lasers with different pulse durations. The fit to the data

show that those pulse durations are 14.7 ps (circles) and 7.6 ps

(squares). These points are indicated on the plot in (c). e Two

identical pulses separated by an appropriate delay can fully transfer

the population. Each pulse has sufficient energy to rotate the state to

the equator of the Bloch sphere. The appropriate delay is approxi-

mately the qubit cycle time 2p/xq. It is slightly smaller due to the off-

axis rotation caused by the Rosen–Zener dynamics. f Data showing

the effect in (e). As the delay between the pulses is scanned, the

transition probability goes from 0 to 1. The maximum is less than one

due to detection errors


123

correspondence between the measured pulse amplitude and

the pulse area h on the x-axis of the plot.

These results show that by controlling the intensity and

delay between two fast pulses, any arbitrary Bloch sphere

rotation can be achieved in tens of picoseconds.

2.2 Weak pulses

In the above section, the pulse area was large, such that a

single pulse had a significant effect on the qubit state. If

instead the area per pulse is small (h� 1), then many pulses

are required to coherently drive the qubit substantially. In

this case, the analysis is better understood in the frequency

domain. The Fourier transform of a train of equally spaced

pulses with a fixed phase relationship is a frequency comb,

with teeth spaced by the repetition frequency xrep. The

width of an individual tooth in an N pulse train scales like

xrep/N. If the width of a tooth is small compared to the tooth

spacing (N � 1), then the comb can be thought of as an

ensemble of CW lasers. All that remains is to ensure that the

frequency comb spectrum includes optical beat notes that

are resonant with the qubit splitting xq.

Note that since the qubit transitions are driven by a fre-

quency difference between comb lines rather than by an

absolute optical frequency, the carrier-envelope phase

(CEP) is therefore irrelevant and does not need to be sta-

bilized. However, in order to coherently drive the qubit, it is

important that the beat note at the qubit splitting be stable.

In general, well-designed mode-locked lasers enjoy excel-

lent passive stability of their repetition rate (comb tooth

spacing) over the time scale of a coherent qubit operation

(microseconds), so that individual operations are coherent.

Over longer times, however, drifts in the repetition rate will

spoil attempts to signal average or concatenate operations.

The fractional drift of the repetition rate, similar to the

fractional linewidth and drift of a free-running CW laser, is

typically in the range of *10-7 over minutes. This drift can

be eliminated by actively stabilizing the laser repetition

rate, using a piezo-mounted end mirror [9].

2.2.1 Single comb

A single comb of equally spaced components can drive

stimulated Raman transitions if the qubit splitting is an

integer multiple of the comb teeth spacing, as shown in

Fig. 6a:

xq ¼ nxrep; n 2 Z ð19Þ

This condition is equivalent to Eq. 16. The Rabi frequency

can be computed by summing the effect of all pairs of

comb teeth separated by xq [9]. For two CW phase-locked

beams with single-photon Rabi frequencies g1 and g2

(assumed to be real), the Raman Rabi frequency between

qubit states is X ¼ g1g2=2D. For an optical frequency

comb resulting from hyperbolic secant pulses, the kth comb

tooth at frequency kxrep from the optical carrier has single-

photon Rabi frequency

gk ¼ g0

ffiffiffiffiffiffiffiffiffiffiffi

xreps2

r

sechðkxrepsÞ; ð20Þ

where g20 ¼

Pþ1k¼�1 g2

k ¼ ð�I=2IsatÞc2. The net two-photon

Rabi frequency from the frequency comb is therefore

X ¼X

þ1

k¼�1

gkgkþn

2Dð21Þ

(a)

(b)

(c)

Fig. 5 Data showing fast phase rotation caused by pair of pulses.

a Ramsey sequence: the frequency of two microwave p/2 pulses is

scanned. In between the microwaves, two fast laser pulses with delay

set to cancel x-rotation are inserted. Fringe shift is then measured as a

function of pulse area. b Data showing fringe shift. Circles No laser

pulses, Squares Laser pulses of pulse area equal to 1.25, showing

phase rotation angle of 0.49. c Measured z-rotation angle as a function

of pulse area

J. Mizrahi et al.

123

� X0sechxqs

2

� �

; ð22Þ

where n is the number of comb teeth spanning the qubit

splitting (Eq. 19), X0 ¼ g20=2D and we assume the beat

notes at xq all add in phase since the pulse has negligible

frequency chirp. The remaining hyperbolic secant factor is

nearly unity when the individual pulse bandwidth 1/s is

much greater than the qubit frequency splitting xq.

This expression can be connected to the time domain

analysis above in a straightforward manner. In Eq. 13, the

number of pulses N can be replaced by time t using the

relation N = 2pxrep t. This shows that the Rabi frequency

is related to the rotation angle u by:

X ¼ 2pxrepu ð23Þ

Equation 13 also shows that full contrast requires

sin2ðu=2Þ ¼ B2, which is equivalent to the condition that

the comb is driving the transition on resonance. This

relation becomes:

sin2 u2

� �

¼ sin2 h2

� �

sech2 xqs2

� �

ð24Þ

) u � hsechxqs

2

� �

ð25Þ

) X ¼ X0sechxqs

2

� �

ð26Þ

The second line follows from the small angle approxima-

tion, and the third line is the second multiplied by 2pxrep.

This shows that the constant X0 ¼ 2pxreph. From this, it is

clear that the approximation made in treating the pulse

train as an ensemble of CW lasers is equivalent to the

assumption that the effect of an individual pulse is small.

In addition to the resonant beat note at the qubit fre-

quency, there will also be many beat notes at integer

multiples of xrep away from the qubit frequency from the

multitude of comb teeth splittings. These other beat notes

will lead to a shift in the qubit resonance and can be

thought of as a higher-order four-photon AC Stark shift.

From Eq. 22, the strength of the beat note at jxrep is

characterized by its resonant Rabi frequency

Xj � X0sechðjxreps=2Þ. The net four-photon Stark shift is

then a sum over all non-resonant beat notes,

d4 ¼ �X

j¼�1

1

j 6¼n

X2j

2ðjxrep � xqÞ

¼ � X20

2xrep

X

j¼�1

1

j 6¼0

sech2½ðjþ nÞxreps=2j

ð27Þ

� 7fð3Þp2

� �

X20xqsxrep

ð28Þ

where f is the Riemann zeta function. The last expression

is valid in the case where xreps� 1 and to lowest order in

xqs/2. For laser pulses of s = 10 ps duration with a rep-

etition rate xrep/2p = 80 MHz and net Rabi frequency

X=2p ¼ 1 MHz, for the 171Yb? qubit we find a resultant

4-photon Stark shift of d4/2p & ?8.5 kHz. It should be

noted that Eq. 28 could also be derived from the time

domain Rosen-Zener solution discussed in Sect. 2.1 [39].

2.2.2 Two combs

Equation 19 requires a laser with a repetition rate that is

commensurate with the qubit splitting. However, this may

be difficult to achieve in practice, and in any case it is

undesirable for non-copropagating laser pulses—such a

laser cannot generate the spin-dependent forces discussed

in Sect. 3 Moreover, the repetition rate on many mode-

locked lasers cannot be easily controlled to stabilize drifts.

It is therefore convenient to generate two combs, with one

frequency shifted relative to the other, typically via an

AOM as shown in Fig. 6b. In this way, Raman transitions

are controlled through this frequency offset and this con-

figuration allows atomic forces to be exerted in a given

direction when tuned near motional sideband transitions

(see Sect. 3). Finally, drifts in the repetition rate can be

measured and fed forward onto a downstream modulator,

in case the repetition rate of a laser is not accessible. This

feed-forward effectively eliminates drift in the relevant

comb beat note to drive qubit transitions by 29, as the

‘‘breathing’’ of the comb teeth is compensated by the offset

comb [40].

Including an offset frequency xA between the two

combs, the condition for driving transitions now becomes:

xq ¼ nxrep � xA; n 2 Z ð29Þ

In order to allow for the possibility of spin-dependent

forces in a counterpropagating geometry, we exclude the

offset frequency values xA = kxrep or (k ? 1/2)xrep,

ωrep

ωqω

Aω

q ωrep

(a) (b)

Fig. 6 a One frequency comb can drive Raman transitions if pairs of

comb lines are separated by the qubit frequency, leading to the

condition in Eq. 19. b Two frequency combs can drive Raman

transitions together if a frequency offset xA between the combs

causes lines from the separate beams to be spaced by the qubit

frequency, leading to the condition in Eq. 29


123

k 2 Z. Figure 7 shows Rabi flopping driven by two offset

optical frequency combs, in a copropagating geometry

where the repetition rate is directly stabilized.

The Rabi frequency for the case of two offset combs is

exactly as written for the case of a single comb (Eq. 22),

where this time g20 ¼ ð�I=2IsatÞc2 characterizes the intensity

�I of each of the two combs. For the offset combs, the four-

photon AC Stark shift is modified from the asymmetry in

the spectrum of two-photon beat notes. Once again sum-

ming over all non-resonant beat notes, we find

d4 ¼�X

j¼�1

1

j 6¼n

X2j

2ðjxrepþxA�xqÞ�X

1

j¼�1

X2j

2ðjxrepþxAþxqÞ

¼ � X20

2xrep

X

j¼�1

1

j6¼0

sech2½ðjþ nÞxreps=2j

2

6

6

4

�X

1

j¼�1

sech2½ðj� nÞxreps=2jþ 2r

#

ð30Þ

X20

2xrep

3:412xqsþ sech2ðxqs=2Þ 1

2rþ 1

1þ 2rþ 1

2r� 1

� � �

ð31Þ

where ~xA ¼ xAðmod xrepÞ; and r � ~xA=xrep is the frac-

tional number of comb teeth that the two combs are offset

(0 \ r\ 1 and r = 0.5), and again we assume xreps � 1.

The extra terms in the Stark shift compared to the single

comb case (Eq. 28) account for the closer asymmetric beat

notes. Interestingly, the net four-photon AC Stark shift can

be nulled by choosing a particular offset frequency for a

given pulse duration. In the 171Yb? system for example, we

find that a value of r * 0.35 (0.40) nulls the Stark shift for

pulse duration s & 5 (10) ps. For infinitesimally short

pulses ðs! 0Þ, the Stark shift vanishes at the value

r ¼ 1=ffiffiffiffiffi

12p

.

3 Entanglement of spin and motion

The above section treated spin flips from copropagating

pulses. Consider now a pair of counterpropagating pulse

trains, as shown in Fig. 8. The pulses are timed such that

they arrive at the ion simultaneously, and the entire train

has effective pulse area of order p. The frequency space

picture is the same as shown in Fig. 6b—the two combs

have a relative frequency shift, such that there exist pairs of

comb lines that match the qubit splitting. However,

absorption from one comb and emission into the other now

leads to momentum transfer. Moreover, the direction of the

momentum transfer is spin-dependent, leading to a spin–

motion coupling. The form taken by that coupling will

differ based on the duration of the pulse train. If the pulse

train is much faster than the trap period, the result will be a

spin-dependent kick: j0i and j1i will receive momentum

kicks in opposite directions. If the pulse train is much

slower than the trap period on the other hand, motional

sidebands will be resolved. In the Lamb-Dicke limit where

the ion motion is confined much smaller than the optical

wavelength, the motion will not be changed when on res-

onance, while a phonon will be added or subtracted when

the beat note between the combs is detuned by the trap

frequency.

To understand this process, first consider the effect of a

single pair of pulses that arrive simultaneously on the ion

from opposite directions. If the two pulses have orthogonal

linear polarizations which are mutually orthogonal to the

quantization axis (lin\lin), then transitions can only be

driven via the polarization gradient created by the two

pulses. The Rabi frequency then acquires a sinusoidal

position dependence. Under the instantaneous pulse

approximation (s = 0), the Hamiltonian for the ion-pulse

interaction becomes:

Fig. 7 Rabi oscillations driven by a pair of copropagating combs

with an AOM shift between them. In these data, the laser repetition

frequency is directly stabilized

mode-locked355 nm laser

pulsepicker

bs

AOM AOM

Fig. 8 Experimental layout for counterpropagating geometry. The

pulse train is split, and a frequency shift between the two arms is

imparted by AOMs

J. Mizrahi et al.

123

H ¼ � h2dðt � t0Þ sin Dkx̂þ /ðt0Þ½ r̂x ð32Þ

where h is again the total pulse area, t0 is the arrival time of

the pulse pair, Dk is the difference in wavevectors, x̂ is the

position operator for the ion, and /(t0) is the phase

difference between the pulses. The time dependence of this

phase difference comes from the AOM frequency shift:

/ðtÞ ¼ xAt þ /0 ð33Þ

where /0 is assumed to be constant over the course of one

experiment. Equation 32 can be directly integrated to obtain

the evolution operator for a single pulse arriving at time t0:

Ut0 ¼ exp �i

Z

HðtÞdt

� �

¼ eih2

sin Dkx̂þ/ðt0Þð Þr̂x

ð34Þ

¼X

1

n¼�1ein/ðt0ÞJnðhÞD½ingr̂n

x ð35Þ

where Jn is the Bessel function of order n, D is the coherent

state displacement operator [41], and g is the Lamb-Dicke

parameter. If we do not assume instantaneous pulses, this

result will be modified slightly. However, it will not change

the qualitative behavior described below. This is shown in

the ‘‘appendix’’.

Equation 35 consists of operators of the form D½ingr̂nx ,

which impart n momentum kicks of size g together with

n spin flips. Physically, this corresponds to the process of

absorbing a photon from one beam, emitting a photon into

the other beam, repeated n times. Each process of

absorption followed by emission changes the momentum

by g. The amplitude for the nth such process is given by the

Bessel function Jn(h), together with a phase factor. The net

action of this operator on a spin state j0i and coherent

motional state jai is therefore to create a superposition of

states of different size kicks, with alternating spin states.

This is shown graphically in Fig. 9b.

This behavior can be understood as the scattering of the

atomic wavepacket off the standing wave of light, known

as Kapitza-Dirac scattering [42–44]. It has been directly

observed in atomic beams [43, 44]. It is also similar to the

behavior observed in d-kicked rotor experiments [45].

However, all of the work cited above dealt only with

atomic motion; the interaction described here is compli-

cated by the presence of the spin operator.

The evolution operator ON for a train of N pulses will

consist of a sequence of operators of the form 35, separated

by free evolution:

ON ¼ UtN . . .UFEðt3 � t2ÞUt2 UFEðt2ÞUt1 ð36Þ

where tn is the arrival time of the nth pulse, and UFE(T) is

the free evolution operator for time T, given by:

UFEðTÞ ¼ e�ixtrapTayae�ixqTr̂z=2 ð37Þ

Let the total pulse train area be given by H, so that a single

pulse area is h ¼ H=N. Assume that N is sufficiently large

such that the single pulse evolution operator in Eq. 35 can

be approximated to first order in 1/N:

Utk � 1þ iH2N

ei/ðtkÞD ig½ þ e�i/ðtkÞD �ig½ � �

r̂x ð38Þ

The free evolution between the pulses can be absorbed into

the pulse operator by transforming to the interaction

picture. Define:

(a)

(b)

(c)

(d)

Fig. 9 Phase space diagrams of pulse action. Red closed circles

indicate j0i, while blue open circles indicate j1i. The size of the circle

indicates the population in that state. a Phase space diagram of an ion

initially in the state j0ijai. b Upon the arrival of a pulse pair, the ion is

diffracted into a superposition of states as in Eq. 35. c After N pulse

pairs satisfying Eq. 47, population coherently accumulates in the state

j1ija� igi and no other state, as in Eq. 46. Similarly, population

initially in j1ijai coherently accumulates in j0ijaþ igi. d Theoretical

error (1-fidelity) of (c) as a function of N. The convergence is very

fast—4 pulses is 96 %, 8 pulses 99.9 %, and 16 pulses 99.99 %


123

q� ¼ xq � xA ð39Þ

In the interaction picture, Ut_k becomes:

Vtk ¼ UyFEðtkÞUtk UFEðtkÞ

¼ 1þ iH2N

ei/0 D igeixtraptk� �

� eiqþtk r̂þ þ eiq�tk r̂��

þ H.c.�

ð40Þ

Under this transformation, the interaction picture pulse

train operator becomes:

~ON ¼Y

1

k¼N

Vtk ð41Þ

There will now be two different approximations made in the

fast regime (xtraptN� 1) and the slow regime (xtraptN� 1).

3.1 Fast regime

In the fast regime, xtrap & 0 during the pulse train, so that

the ion is effectively frozen in place. Equation 40 then

becomes:

Vtk ¼ 1þ iH2N

ei/0 D ig½ eiqþtk r̂þ þ eiq�tk r̂��

þ H.c.� �

ð42Þ

The net pulse train operator for a series of fast pulses (Eq.

41) will then be a product of the Vtk in Eq. 42. In general,

this product will be extremely complicated. When

expanded, the coefficients of each spin/motion operator

will consist of terms of the formP

eiq�tk . Typically the

norm of such sums is [1. For large N, the 1/N suppression

will result in ~ON � 1, i.e., the pulses do nothing. However,

suppose qþtk=2p 2 Z for all pulses, while q�tk=2p 62 Z.

The pulses are then resonant—in frequency space, this is

equivalent to satisfying the bottom sign resonance condition

in Eq. 29, but not the top sign. The q? terms in the product

in Eq. 41 will then coherently add, while the q- terms will

not. As the number of pulses grows, the non-resonant terms

are strongly suppressed and can be discarded. In frequency

space, this is equivalent to the statement that the comb lines

narrow with increasing N, resulting in decreased amplitude

for non-resonant processes. Equation 42 therefore becomes:

Vtk ¼ 1þ iH2N

ei/0 D ig½ r̂þ þ e�i/0 D �ig½ r̂��

ð43Þ

The pulse train operator is now a product of identical

operators:

~ON ¼ 1þ iH2N


� �N

�!N!1exp

iH2


� �

ð44Þ

¼ cosH2þ i sin

H2


ð45Þ

For a total pulse area of H ¼ p, Eq. 45 becomes:

~O ¼ iei/0 D ig½ r̂þ þ ie�i/0 D �ig½ r̂� ð46Þ

This is the desired spin-dependent kick operator. In

summary, if the following conditions are satisfied:

qþtk

2p2 Z ð47Þ

q�tk

2p62 Z ð48Þ

then in the limit N !1, a pulse train will create a spin-

dependent kick in which j0i is kicked up and j1i is kicked

down. Alternatively, we could have chosen q�tk=2p 2 Z

and qþtk=2p 62 Z, in which case the final kick directions

would be reversed. In essence, Eq. 47 says that for every

pulse, the phase accumulated due to qubit precession (xqtk)

plus that accumulated due to the RF modulation on the

AOMs (xAtk) should be a multiple of 2p. If the pulses are

equally spaced, then tk = 2p k/xrep, and Eq. 47 is equiv-

alent to Eq. 29. It is also important to note that this spin-

dependent kick does not depend on being in the Lamb-

Dicke regime.

The condition that only one of q? or q- is resonant is

equivalent to the condition that the hyperfine frequency not

be an integer or half-integer multiple of the repetition rate:

xq 6¼nxrep

2; n 2 Z ð49Þ

If xq/xrep is a half-integer or integer, then it is impossible

to be resonant with a kick in one direction without also

being resonant with a kick in the opposite direction. The

net result will be that the kicks will cancel, and the pulse

train will not drive any transitions at all.

It is clear from the time domain analysis that these

pulses do not have to be equally spaced. Indeed, numerical

optimization shows that the best SDK fidelity is achieved

for unequally spaced pulses. To understand this result,

consider the product in Eq. 41 to lowest order in H=N:

~ON ¼ 1þ iH2N

ei/0 D½igX

N

k¼1

eiqþtk

!

r̂þ þ H.c.

" #(

þ ei/0 D½igX

N

k¼1

eiq�tk

!

r̂� þ H.c.

" #)

þ O H=Nð Þ2� �

ð50Þ

When the resonance condition in Eq. 47 is satisfied, then

the coefficientsPN

k¼1 e�iqþtk ¼ N, and the top term in

brackets generates the spin-dependent kick. The bottom

term, corresponding to the ‘‘wrong way’’ kick, will lead to

infidelity in the SDK. Above it was stated that the

J. Mizrahi et al.

123

coefficientPN

k¼1 e�iq�tk is O(1) when off resonance,

leading to strong suppression for large N. However,

maximal suppression occurs when this term is zero. This

leads to the condition:

X

N

k¼1

eiq�tk ¼ 0 ð51Þ

Note that this is a second condition on the arrival times tk(the first condition being the ordinary resonance condition

in Eq. 47). Finding tk which satisfy both of these conditions

will result in a significantly improved SDK fidelity, as

compared to satisfying resonance only. However, in gen-

eral an equally spaced pulse train will not satisfy Eq. 51.

This is why using an unequally spaced pulse train can

allow higher SDK fidelity—it allows satisfying both Eq. 47

and Eq. 51.

Note that even with satisfying both the conditions

mentioned, the SDK fidelity will still not be 1, as Eq. 50 is

only to first order in H=N. Higher-order terms will lead to

infidelity. These terms will also have conditions under

which they are zero. By numerically optimizing the arrival

times tk, the SDK infidelity can be suppressed to a very

high order with only a small number of pulses. Ultimately,

the degree to which the unwanted terms can be suppressed

is limited by the number of degrees of freedom available in

choosing the pulse arrival times.

Figure 9d shows the numerically optimized fidelity for

different numbers of pulses, allowing unequal spacings.

Because the pulse train is generated with delay lines (see

below), the 8 pulse train has only 3 free parameters.

Nevertheless, simulations show that a fidelity better than

99.9 % is achievable after only 8 pulses. With 16 pulses,

the fidelity can be better than 99.99 %. Here the AOM

difference frequency is fixed. Higher AOM frequencies

could potentially allow even higher fidelities.

In order for the approximation xtrap & 0 to be valid, the

duration of the pulse train must be at least 2–3 orders of

magnitude shorter than the trap period. A typical trap

period if of order 1ls, meaning the pulse train cannot be

longer than a few nanoseconds. However, the repetition

rate of pulses produced by the available lasers is only

80–120 MHz. At that rate, the ion would experience sig-

nificant trap evolution even over the course of a small

number of pulses. As an alternative, a single pulse from the

laser followed by a sequence of delay lines can create a

very fast pulse train, as shown in Fig. 10. The limitation on

the speed is then determined by the AOM frequency.

We demonstrated in [11] the creation of a spin-depen-

dent kick of the form in Eq. 46. There, we showed that such

kicks entangle the spin with the motion, while a second

kick can disentangle the motion at integer multiples of the

trap period.

Direct observation of the motional state of a trapped ion

is extremely difficult, and motional information is typically

determined by mapping to the spin [46]. Therefore, to

detect that we created the operator in Eq. 46, it is necessary

to infer the motional entanglement from its impact on the

measured spin state. To do this, we performed a Ramsey

experiment using microwaves. The experimental sequence

was to (1) Initialize the spin state to j0i, (2) Perform a p/2

rotation using near resonant microwaves, (3) Perform a

spin-dependent kick using a single pulse through the

interferometers, (4) Wait a time Tdelay (5) Perform a second

spin-dependent kick, (6) Perform a second p/2 rotation, and

(7) measure the state of the ion. The frequency of the

microwaves was then scanned. If the motion is disentan-

gled from the spin, the result should be full contrast of the

Ramsey fringe. On the other hand, if the spin and motion

are entangled, then the trace over the motion will destroy

the phase coherence. The result will be no Ramsey fringes.

The motion should disentangle when Tdelay matches an

integer multiple of the trap frequency.

Figure 11 shows the results of this experiment. Plotted is

the Ramsey contrast as a function of Tdelay. The clear

collapse and revival of contrast is a strong indicator that the

pulses are indeed performing the spin-dependent kick in

Eq. 46. This sort of interferometry is similar to that dis-

cussed in [34].

3.2 Slow regime

In the slow regime, the pulse train is much longer than the

trap cycle time: tN� 1/xtrap. Now assume that the ion is in

the Lamb-Dicke regime: gffiffiffiffiffiffiffiffiffiffiffi

�nþ 1p

� 1. In this regime, the

following approximation can be made:

D igeixtraptk�

� 1þ ig eixtraptk ay þ e�ixtraptk a� �

ð52Þ

where a and ay are the harmonic oscillator annhilation and

creation operators. Substituting this approximation into

Eq. 40 yields:

Fig. 10 Optical layout for creating fast pulse train from a single pulse


123

Vtk ¼ 1þ iH2N� ei/0 1þ ig eixtraptk ay þ e�ixtraptk a

� ��

� eiqþtk r̂þ þ eiq�tk r̂��

þ H.c.�

ð53Þ

There are now six phases to consider, associated with six

different operators: eiq�tk ; eiðq�þxtrapÞtk ; and eiðq��xtrapÞtk . The

situation is then similar to the strong pulse regime: If one of

these phases satisfies resonance (i.e., equal to 1 for all tk)

while the others do not, then the other terms will be

negligible in the limit of large numbers of pulses. For

example, suppose that ðqþ þ xtrapÞ=2p 2 Z; while none of

the other phase terms satisfy this condition. In that case,

Eq. 53 becomes:

Vtk ¼ 1þ iHg2N

iei/0 ayr̂þ � ie�i/0 ar̂��

ð54Þ

As in the fast regime, the pulse train operator in Eq. 41 is

now the product of identical operators and converges to:

~O ¼ cosHg2þ i sin

Hg2

iei/0 ayr̂þ � ie�i/0 ar̂��

ð55Þ

This is Rabi flopping on the blue sideband. Similarly, the

other resonance conditions correspond to red sideband and

carrier operations. This behavior is shown in Fig. 12a.

We previously reported in [9] on using pulse trains to do

resolved sideband operations, as described above. There we

demonstrated sideband cooling and two ion entanglement

using the Mølmer-Sørensen technique [16, 28, 47].

Figure 12 is experimental data showing the crossover

between the slow and fast regimes. In these data, the

transition probability was measured as a function of AOM

detuning. In (a), sideband features are clearly resolved. The

peaks correspond to the carrier and sidebands at each of the

three trap frequencies (1.0, 0.9, 0.1) MHz. These transitions

follow from Eq. 53. As the power is increased and the pulse

train duration decreased, the sidebands become less

resolved, as the behavior crosses over from the slow regime

to the fast regime. In (e), all of this structure has been

(a)

(b)

(c)

(d) (e) (f)

Fig. 11 (Reproduced from [11]). a Ramsey experiment to measure

effect of spin-dependent kicks. Two spin-dependent kicks separated

by a time T are placed between two microwave p/2 pulses. b Ramsey

contrast as a function of delay between kicks. Clear revivals of

contrast are seen at integer multiples of the trap period. c Close up of

one revival peak. The small modulation present in the peak is due to

uncompensated micromotion. The width of the peak is a function of

the ion temperature and the micromotion amplitude. d–f Phase space

representation at various points on the plot in (a). Also shown are the

Ramsey frequency scans at those points, showing the presence or lack

of contrast

(a)

(b)

(c)

(d)

(e)

Fig. 12 Data showing the crossover between the slow, resolved

sideband regime and the fast, impulsive regime. Each plot corre-

sponds to scanning the frequency of an AOM in one of the arms of

counterpropagating pulse trains. a X� xt, and sideband transitions

are clearly resolved. b–d As the pulse train power is turned up and the

Rabi frequency increases, the lines begin to blur together. e No

features are resolved at all, meaning all sidebands are being driven

J. Mizrahi et al.

123

washed out, and the motional transition is now described

by impulsive kicks. From a sideband perspective, all

sidebands are being driven simultaneously.

4 Ultrafast gates

The goal of creating spin-dependent kicks of the form in

Eq. 46 is to execute a fast two ion entangling gate. Such a gate

would not be based on sidebands and would therefore be

fundamentally different from previously implemented two ion

gates. Because it does not depend on addressing sidebands,

such a gate will be temperature insensitive and would not

require the ion to be cooled to the motional ground state or

even cooled to the Lamb-Dicke regime. Additionally, the

Raman lasers generating the spin-dependent kick can be

focused down to address just two adjacent ions in a long chain.

If the gate is sufficiently fast, the other ions do not participate

in the interaction. In principle, this allows this type of gate to

be highly scalable. There have been theoretical proposals for

such a gate in [48] and in [49]. Both schemes rely on using a

sequence of spin-dependent kicks, timed such that the col-

lective motion returns to its original state at the end of the

process. This leaves a spin-dependent phase.

To understand the origin of this spin-dependent phase,

consider a simple sequence of three spin-dependent kicks

applied to two ions:

1. t = 0: momentum kick of size þDk

2. t = T: momentum kick of size �2Dk

3. t = 2T: momentum kick of size þDk

This is a simplified version of the scheme proposed by

Duan [49]. Suppose that the total length of the kicking

sequence is much faster than the trap period: xtrapT� 1. In

that case, trap evolution during the kicks can be ignored,

and the ions behave as free particles. The first kick imparts

a momentum to each ion of Dk. The ions then move at a

constant velocity away from equilibrium, until the second

kick reverses the direction. The third kick then stops the

motion of the ions at (nearly) the original position.

For two ions, there are four different possible spin states.

Each will have a different motional excitation in response

to these kicks, as shown in Fig. 13a.

If the ion spin state is j0ij0i or j1ij1i, the two ion energy

from the Coulomb interaction does not change during the

sequence. However, for j0ij1i and j1ij0i, the energy

changes as the ions get further apart and then closer

together. The time-dependent energy difference between

these two configurations is:

DEðtÞ ¼ e2

d� e2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

d2 þ dðtÞ2q � 2e2dðtÞ2

d3ð56Þ

where e is the electron charge, d is the distance between the

ions in equilibrium, and d(t) is the displacement of each ion

from equilibrium as a function of time (see Fig. 13b). The

acquired phase difference from this process is given by:

D/ ¼Z

2T

0

DEðtÞdt ¼ 4e2Dk2T3

3d3m2ð57Þ

We see then that the motional state (nearly) returns to its

original state at the end of the process, while j0ij1i and

j1ij0i acquire a phase relative to j0ij0i and j1ij1i. This is

thus a phase gate. Note that the motion is entirely driven.

Equation 57 is valid only because the ions are effectively

free particles. The natural harmonic motion in the trap does

not lead to phase accumulation.

The fidelity of the phase gate described above is limited

by free evolution in the trap. Because the gate is not truly

instantaneous, there will be a small amount of residual

entanglement with the motion at the end of the process.

This infidelity can be eliminated by more complex kicking

sequences, described below.

Alternatively, this process can be viewed as exciting the

two normal modes of motion in the trap. Phase space

diagrams of the kick sequence are shown in Fig. 14 for the

two different modes (center of mass and relative), both in

the non-rotating frame and in the rotating frame. We can

determine the evolution of a coherent state jai subjected to

(a)

(b)

Fig. 13 a The ground state of the motion is excited into four different

possible configurations depending on the two ion spin state. The

dashed circles show the original, equilibrium position of the ions. The

arrow and solid circles show the path followed after the first kick.

b In the limit where the kicks are much faster than the trap period, the

trap evolution during the kicking sequence is negligible, and the ions

can be considered as free particles. The displacement d of each ion

from equilibrium as a function of time is shown


123

the kicks described above. For simplicity in this example,

we will treat the ground state a = 0.

At the end of the simple pulse sequence, the state of the

ions in a normal mode of frequency x is:

eig2ð�4 sinðxTÞþsinð2xTÞÞjigð1þ ð�2þ e�ixTÞe�ixTÞi ð58Þ

The phase for a given mode is given by:

/ � � 2Dk2T

m1þ x2T2

3

� �

ð59Þ

The phase difference between the two modes is thus given

by:

D/ ¼ � 2Dk2T3

3mx2

R � x2C

� �

ð60Þ

¼ 4e2Dk2T3

3d3m2ð61Þ

where xC and xR are the frequencies for the center of mass

and relative motion modes. This is the same expression

found using the Coulomb picture in Eq. 57.

The phase difference in Eq. 57 can also be extracted by

examining the phase space area enclosed by this sequence.

The trajectories in the rotating frame are shown in Fig. 14c,

d. In the rotating frame all paths are driven, which leads to

phase accumulation. If a coherent state is driven through a

trajectory which encloses an area A in the rotating frame

phase space, that coherent state acquires a phase 2A [50,

51]. This fact allows us to determine the phase acquired

simply by calculating the area enclosed in Fig. 14c and d.

This calculation once again matches the phase in Eq. 59.

It is worth pointing out that although the simple example

illustrated in Fig. 13 uses the transverse modes of motion,

such a phase gate also works with the axial modes of

motion. Moreover, if the axial modes of motion are used,

the displacement d is directly along the line separating the

two ions, resulting in a larger modification of the Coulomb

interaction. Equation 61 applies equally for axial or

transverse modes. For transverse modes, the term in

parentheses is xz2, while for axial modes it is 2 xz

2. So if all

other parameters are held constant there is a factor of 2

greater phase when using axial modes instead of transverse.

However, there is added flexibility in using transverse

modes, as will be discussed below.

Unfortunately, this simple sequence of kicks has two

serious limitations. First, the phase obtained from this

sequence is small. Plugging realistic experimental param-

eters (d = 5 lm, T=100 ns, Dk ¼ 2� ð 2p355 nmÞ into Eq. 57

we find a phase difference of p /780, significantly smaller

than the p/2 needed for a maximally entangling phase gate.

Second, the motion does not factor completely at the end of

the pulse sequence, but some residual entanglement

remains. This is clearly seen in Eq. 58 where the final state

now depends on g; x, and T. Both of these limitations can,

in principle, be overcome by using more complicated pulse

sequences with many laser pulses strung together to give a

larger momentum kick.

The theory proposals in [48] and [49] both go beyond

the simple pulse sequence presented above. In [49], Duan

solves these problems by using many pulses in quick

succession. Moreover, he shows that with more compli-

cated pulse sequences the errors can be reduced while still

completing the gate in a time much faster than the trap

period. This allows the scheme to be used on a pair of

adjacent ions in a long chain. If the gate is sufficiently fast,

the other ions are not disturbed and the gate is scalable to

large ion crystals. Unfortunately, this scheme relies on a

very large number of pulses ([1,000) in a very short period

of time (\5 ns), and there is not currently a commercial

laser available with high enough power and fast enough

repetition rate to implement this scheme in our system.

In Garcia-Ripoll [48], the trap evolution is used to

control the trajectory in phase space. By correctly choosing

the timing of a series of spin-dependent kicks, the relative

phase accumulated by the two normal modes can be con-

trolled and both phase space trajectories can be closed,

returning the ions to their original position. Here we will

present an experimentally achievable extension of their

scheme which could perform an entangling phase gate on

(a) (b)

(c) (d)

Fig. 14 Phase space picture of the kick sequence described in the

text. a, b are shown in the non-rotating frame, where free evolution

follows circles in phase space. c, d are in the rotating frame. The

phase difference is given by twice the difference in the enclosed area

J. Mizrahi et al.

123

two ions in approximately 1 ls. There have been other

proposals of experimentally achievable timings for fast

entangling gates, see Bentley [52].

For simplicity, we choose a scheme similar to that in

Garcia-Ripoll [48], but to accumulate more phase we

replace each of the four spin-dependent-kicks in [48] with

7 spin-dependent kicks. The gate will be executed on

transverse motional modes. The advantage of using trans-

verse modes lies in the fact that the frequency ratio

between the center of mass and relative modes of motion

can be tuned. For the axial modes, that ratio is fixed atffiffiffi

3p

.

The transverse center of mass frequency for the gate

described here is 1.67 MHz; the relative mode frequency is

1.48 MHz. Experimentally each kick is derived from a

single pulse of a mode-locked laser with a repetition rate of

80.16 MHz, so the delay between successive kicks is

12.5 ns. This is not negligible compared to the trap period

of 599 ns. As a result, the trap evolution between the kicks

is important and must be taken into account. We apply 7

spin-dependent kicks with 7 successive pulses from the

laser. Because each kick also flips the spin of the ion, the

direction of the spin-dependent kick must be reversed

between successive pulses to continue to add momentum to

the system. After the 7 spin-dependent kicks, the ion is

allowed to evolve in the trap for a time t1 = 263 ns, and

then 7 more spin-dependent kicks are applied in the same

direction. The system then evolves freely for a time

t2 = 188 ns. The first three steps are then reversed: 7 more

kicks in the opposite direction, evolve for t1, 7 more kicks

to return the system to its original location. The total gate

time is 1.02 ls. Figure 15 shows the path in phase space

for both the center of mass and relative modes.

The scheme presented in Fig. 15 is just one of many

possible ways to perform this phase gate. There are three

constraints. Both the normal mode phase spaces must

close, and the differential phase between the two phase

spaces must be p/2. As an added constraint, all the kicks

must be spaced by multiples of the fixed laser repetition

rate. In a span of 1 ls, the laser produces 80 pulses. Each

pulse can give no momentum kick or a momentum kick of

g in either direction. This means that there are

380 = 1.5 9 1038 different possible pulse sequences. Most

of those do not fulfill the constraints above, but a detailed

search reveals that there is a very large set of solutions.

However, it is unlikely that there exist solutions much

faster than a few hundred nanoseconds using the experi-

mental system presented here, as the algorithm is ulti-

mately limited by the repetition rate of the laser.

5 Conclusion

We have demonstrated that mode-locked lasers are an

extremely versatile tool in the coherent control and

entanglement of trapped ions in both the fast and slow

regimes.

In the slow regime, we have shown that the spectral

features of the frequency comb can be used in much the

same way as CW lasers, where ion-ion entanglement is

produced by addressing sideband transitions. The advan-

tages in this regime are twofold: First, the available power

enables operating much further from resonance, which

reduces laser-induced decoherence. Second, the broad

spectrum allows direct coupling of the qubit levels using a

single beam, without the experimental difficulties associ-

ated with creating a microwave beat note between two CW

beams.

In the fast regime, we have shown that it is possible to

drive arbitrary rotations of a trapped ion in tens of pico-

seconds, which is many orders of magnitude faster than the

coherence time. We have also shown the ability to perform

fast spin-dependent kicks, which opens the door to per-

forming very fast gates. The advantage of these gates is

their insensitivity to temperature, their extreme speed, and

their potential for scalability.

Acknowledgments This work is supported by grants from the US

Army Research Office with funding from the DARPA OLE program,

(a)

(c)

(b)

Fig. 15 Phase space picture of an experimentally realizable phase

gate. a Center of mass mode, b relative motion mode. In the rotating

frame the direction of the spin-dependent kick rotates at the normal

mode frequency. c Depiction of kick sequence. The ion is kicked 7

times by 7 successive laser pulses with 12.5 ns of trap evolution

between each kick. The ion then evolves in the trap for t1 = 263 ns

then kicked 7 more times. After a wait of t2 = 188 ns, the sequence is

reversed with 7 kicks in the opposite direction, free evolution for t1and a final 7 kicks to return the ions to their original position


123

IARPA, and the MURI program; the NSF PIF Program; the NSF

Physics Frontier Center at JQI; and the European Commission AQ-

UTE program.

Appendix: Motional evolution operator with nonzero

pulse duration

In Sect. 3, Eq. 35 was derived by approximating the pulse

as a d-function. This section examines the validity of that

approximation. The pulse duration is of order 10 ps,

meaning it is several orders of magnitude faster than the

trap frequency or the AOM frequency. Therefore, the

Rosen-Zener solution in Sect. 2 can be used, with h!h sin Dkx̂þ /ð Þ in Eqs. 5 and 6:

A ¼ C2 nð ÞC n� h

2p sin Dkx̂þ /ð Þ� �

C nþ h2p sin Dkx̂þ /ð Þ

� �

ð62Þ

B ¼ � sinh2

sin Dkx̂þ /ð Þ� �

sechxqTp

2

� �

ð63Þ

The r̂x term in part of Eq. 4 is given by iB, which can be

expanded using the Jacobi-Anger expansion as:

iB ¼ sechxqTp

2

� �

X

1

oddn¼�1ein/JnðhÞD ing½ ð64Þ

This is nearly identical to the r̂x term in Eq. 35, but with an

overall sech xqTp=2� �

term modifying the populations. The

even-order diffraction terms are of order h2 or higher,

which were assumed to be negligible in Sect. 3. Nonzero

pulse duration can thus be accounted for by replacing

h! hsech xqTp=2� �

. This will correspond to a slight

reduction in the effective pulse area as compared to a

d-function pulse.

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